Symmetry
Why This Matters
Look at a butterfly. The left wing looks just like the right wing. Look at your own face in a mirror. The left side matches the right side. Look at a rangoli or a flower. Turn it a little, and it still looks the same.
This “matching” is called symmetry. It is everywhere — in nature, in art, and in the things people build. A butterfly, the Taj Mahal, a wheel, a snowflake, a kite, a fan. They all feel balanced and pretty. That nice, balanced feeling comes from symmetry.
In this chapter we will learn two simple kinds of symmetry. First, line symmetry — when one half of a shape is the mirror copy of the other half. Second, rotational symmetry — when a shape looks the same after you turn it a little. By the end, you will be able to look at any shape or letter and say how symmetric it is.
The Big Idea
A shape has line symmetry if you can draw a line through it that folds it into two halves that match exactly — like a mirror. That line is called the line of symmetry. A shape has rotational symmetry if you can turn it part of the way around a fixed point and it looks exactly the same as before. Some shapes have many lines of symmetry, some have just one, and some have none at all.
Let’s Break It Down
Line of symmetry — the mirror line
Imagine you draw a butterfly on paper. Now you fold the paper down the middle. If the two halves cover each other perfectly, with no part sticking out, then the fold line is special.
A line of symmetry is a line that cuts a shape into two halves that match exactly when you fold along it. The two halves are mirror copies of each other. That is why a line of symmetry is also called a mirror line.
Here is an easy way to picture it. The line of symmetry works like a mirror standing on the line. Whatever is on the left is reflected to the right, exactly. Figure 9.1 shows this with a butterfly.
Because the two halves match like a mirror image, line symmetry is also called reflection symmetry. “Reflection” just means the mirror image.
Not every line through a shape is a line of symmetry. The fold must make the two halves match exactly. If even a small bit does not match, it is not a line of symmetry.
You fold a shape along a line. The left half sticks out a little past the right half. Is that line a line of symmetry?
No. For a line of symmetry, the two halves must match exactly with nothing sticking out. Since the left half sticks out, the fold does not match, so it is not a line of symmetry.
Finding lines of symmetry in shapes and letters
A shape can have one line of symmetry, many lines, or none at all. Let’s see how to count them.
The trick is to ask: in how many different ways can I fold this shape so the two halves match? Each such fold is one line of symmetry.
Let’s start with letters, because you see them every day. Figure 9.2 shows five capital letters.
Notice three things from Figure 9.2:
- A folds nicely down the middle (left matches right), so it has 1 line.
- H folds two ways — top-to-bottom and left-to-right — so it has 2 lines.
- F cannot be folded into matching halves in any way, so it has 0 lines.
Now the important question: why does a square have 4 lines of symmetry, but a rectangle only 2?
Let’s think about it slowly. A square has all four sides equal. A rectangle has two long sides and two short sides — its sides are not all equal.
Take a square first. You can fold it:
- straight down the middle (top to bottom),
- straight across the middle (left to right),
- along one diagonal (corner to corner),
- along the other diagonal.
All four folds make the two halves match. So a square has 4 lines of symmetry.
Now take a rectangle. The two straight folds (down the middle and across the middle) still work fine. But what about the diagonal? When you fold a rectangle along its diagonal, the long side tries to land on the short side. They are different lengths, so they do not match. The two halves do not cover each other. So the diagonal of a rectangle is not a line of symmetry. That leaves only 2 lines. Figure 9.3 shows this clearly.
So the rule is simple: equal sides give more lines of symmetry. The more “balanced” a shape is, the more lines it has.
Some shapes have no line of symmetry. For example, the letter F, or a shape like a scalene triangle (a triangle with all three sides different). No fold makes their halves match.
Let’s also bring back a shape you met before, so it is fresh.
Here is how triangles do:
- An equilateral triangle (all sides equal) has 3 lines of symmetry.
- An isosceles triangle (2 sides equal) has 1 line of symmetry.
- A scalene triangle (no sides equal) has 0 lines of symmetry.
Why does an equilateral triangle have more lines of symmetry than an isosceles triangle?
An equilateral triangle has all three sides equal, so it can be folded to match in three different ways — one through each corner. An isosceles triangle has only two equal sides, so only one fold (through the top corner) makes the halves match. More equal sides means more lines of symmetry.
Rotational symmetry — same look after a turn
There is a second kind of symmetry. This one is not about folding. It is about turning (also called rotating).
Look at a paper windmill, or a fan. You cannot always fold them into matching halves. But if you spin them a little around the centre, they look exactly the same as before. This is called rotational symmetry.
A shape has rotational symmetry if you can turn it part of the way around a fixed point and it looks exactly the same as it did at the start.
Let’s see this with a square. Put your finger on the centre of a square and turn it by a quarter turn — that is 90°. The square looks exactly the same as before! Figure 9.4 shows what happens to the corners.
The corners swapped places, but the shape is exactly the same. You could not tell it had been turned. That is rotational symmetry.
Now, full warning: every shape looks the same after a full turn of 360°, because a full turn brings it right back to where it started. So a full turn never counts as anything special. The interesting question is: does the shape look the same after a turn smaller than a full turn? If yes, it has rotational symmetry.
Centre, angle and order of rotation
Three words help us describe rotational symmetry. Let’s learn them one at a time, with the windmill in mind.
The centre of rotation is the fixed point you turn the shape around. For a windmill, it is the pin in the middle. For a square, it is the exact centre.
The angle of rotation (or angle of symmetry) is how much you turn the shape to make it look the same again. For the square above, the smallest such turn is 90°.
The order of rotational symmetry is the number of times the shape looks the same as you turn it through one full turn (one full 360°). The square looks the same at 90°, 180°, 270° and at 360°. That is 4 times. So the order of the square is 4.
Here is a handy shortcut to find the order. The shape matches itself at every multiple of its smallest angle, all the way up to 360°. So:
Order = 360° ÷ (smallest angle of rotation)
Example: a square’s smallest angle is 90°, so order = 360° ÷ 90° = 4.
Figure 9.5 shows a pinwheel turning, and how we get its order.
Let’s practise finding the order with a clear example.
We want to find the order of rotational symmetry of an equilateral triangle (all three sides equal). Watch each step.
Find the order of rotational symmetry of an equilateral triangle (all sides and angles equal).
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First, find the centre of rotation. For a balanced shape like an equilateral triangle, it is the exact middle of the triangle. We will turn the triangle around this point.
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Now find the smallest turn that makes it look the same. An equilateral triangle has 3 equal corners spread evenly around the centre. Turning it so one corner takes the next corner’s place means turning by 360° ÷ 3 = 120°. After a 120° turn, the triangle looks exactly the same.
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Now count how many times it matches in one full turn. It matches at 120°, then again at 240° (that is 120° + 120°), and again at 360° (a full turn). That is 3 matching positions.
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So the order of rotational symmetry of an equilateral triangle is 3. (Check with the shortcut: 360° ÷ 120° = 3. It agrees.)
Let’s put the two kinds of symmetry side by side so the difference is clear.
| Feature | Line symmetry | Rotational symmetry |
|---|---|---|
| What you do | Fold the shape along a line | Turn the shape around a point |
| When it works | The two halves match exactly | The shape looks the same after a part-turn |
| Also called | Reflection (mirror) symmetry | — |
| What we count | Number of lines of symmetry | Order of rotational symmetry |
| Key word | Line of symmetry (mirror line) | Centre, angle and order of rotation |
A shape can have one kind, both kinds, or neither. A square has both: 4 lines of symmetry and rotational symmetry of order 4. The letter S has rotational symmetry (turn it 180° and it looks the same) but it has no line of symmetry. So the two kinds are different ideas.
Common Mistakes
A rectangle has 4 lines of symmetry, just like a square.
A rectangle has 4 corners and 4 sides and looks neat and balanced, so it feels like it should fold every way that a square does, including along its diagonals.
A rectangle has only 2 lines of symmetry — one down the middle and one across the middle. Its diagonals are NOT lines of symmetry, because the long side and short side have different lengths and do not match when folded along a diagonal. Only a square (with all sides equal) has 4 lines.
When counting the order of rotation, you also count the starting position, before any turn.
The shape clearly looks the same at the start, so it feels natural to count that 'do-nothing' position as one of the matches.
We count the matches as the shape turns through one full circle. The full turn of 360° brings it back to the start and is counted as the last match — so the starting position is already included as that 360° match. Do not count the start separately, or you will get one too many.
Every shape with rotational symmetry also has line symmetry, and the other way round.
A square has both kinds, and many pretty shapes have both, so it seems like the two always come together.
The two are different. The letter S has rotational symmetry (turn it 180° and it looks the same) but no line of symmetry. The letter A has 1 line of symmetry but no rotational symmetry. A shape can have one without the other.
Quick Check
How many lines of symmetry does a square have?
A square has 4 lines of symmetry: one vertical, one horizontal, and two diagonals. Because all four sides are equal, every one of these folds makes the halves match.
A shape looks the same after a turn of 90°. What is its order of rotational symmetry?
Order = 360° ÷ smallest angle = 360° ÷ 90° = 4. The shape matches itself at 90°, 180°, 270° and 360°.
Which letter has NO line of symmetry?
There is no way to fold F into two matching halves. A folds down the middle, H folds two ways, and O is round with many lines. F has none.
What is the fixed point you turn a shape around called?
The fixed point a shape is turned around is the centre of rotation. The amount of the turn is the angle of rotation, and the number of matches in a full turn is the order.
Practice Problems
Easy
How many lines of symmetry does the letter H have? Name them.
The letter H has 2 lines of symmetry.
One is vertical (straight down the middle) — the left side matches the right side. One is horizontal (straight across the middle) — the top matches the bottom. So H has 2 lines of symmetry.
A shape's smallest angle of rotation is 60°. What is its order of rotational symmetry?
Use the shortcut: order = 360° ÷ smallest angle.
Order = 360° ÷ 60° = 6. The shape matches itself 6 times in one full turn (at 60°, 120°, 180°, 240°, 300° and 360°).
Medium
How many lines of symmetry does an equilateral triangle have, and what is its order of rotational symmetry?
An equilateral triangle has all three sides equal, so it is very balanced.
Lines of symmetry: 3. You can fold it through each corner to the middle of the opposite side, and the halves match. That gives one line per corner, so 3 lines.
Order of rotational symmetry: 3. Its smallest angle of rotation is 360° ÷ 3 = 120°. It matches itself at 120°, 240° and 360°. So the order is 3.
Draw the lines of symmetry of a rectangle and explain why it has only 2.
A rectangle has 2 lines of symmetry: one vertical line down the middle, and one horizontal line across the middle.
Why not the diagonals? When you fold a rectangle along a diagonal, the long side has to land on the short side. They are different lengths, so they do not match. The halves do not cover each other. So a diagonal is not a line of symmetry. That leaves only the 2 straight lines.
Challenge
The letter S has no line of symmetry. But does it have rotational symmetry? If yes, what is its order?
Try folding S any way — left-right or top-bottom. The halves never match. So S has no line of symmetry. That part is true.
Now try turning it. Put your finger on the centre of the S and turn it by 180° (a half turn). It looks exactly the same! So S does have rotational symmetry.
How many times does it match in a full turn? Only at 180° and at 360°. That is 2 times. So the order of rotational symmetry is 2.
This shows that a shape can have rotational symmetry even with no line of symmetry. The two kinds are different.
A circle is special. How many lines of symmetry does it have, and what about its rotational symmetry?
A circle has endless (infinitely many) lines of symmetry. Any straight line that passes through the centre folds the circle into two matching halves. Since you can draw such a line in every direction, there are countless lines of symmetry.
For rotational symmetry, turn the circle by any angle at all around its centre — it always looks exactly the same. So every angle is an angle of rotation. The circle is the most symmetric shape of all.
Summary
- Line of symmetry (mirror line): a line that folds a shape into two halves that match exactly. This is also called reflection symmetry.
- A shape can have one line of symmetry, many lines, or none at all.
- More equal sides means more lines of symmetry. A square (all sides equal) has 4 lines; a rectangle (sides not all equal) has only 2, because its diagonals do not fold into matching halves.
- An equilateral triangle has 3 lines, an isosceles triangle has 1, and a scalene triangle has 0.
- Rotational symmetry: a shape has it if you can turn it part of the way around a fixed point and it looks exactly the same.
- The fixed point is the centre of rotation; the smallest turn that matches is the angle of rotation; the number of matches in one full turn is the order of rotational symmetry.
- Order = 360° ÷ smallest angle of rotation. A full turn of 360° always counts as a match, so it is already included — do not count the starting position twice.
- Line symmetry and rotational symmetry are different. A shape can have one, both, or neither (for example, the letter S has rotational symmetry but no line of symmetry).
What’s Next
You have now learnt how shapes balance — by folding and by turning. Next, in Chapter 10 — The Other Side of Zero, you will meet a brand new set of numbers: the negative numbers, the numbers that live on the other side of zero. They help us talk about things below zero — like cold temperatures, going down in a lift, or owing money.
Frequently Asked Questions
What is a line of symmetry and how do I find it?
A line of symmetry is an imaginary line you can draw through a shape so that one half is the exact mirror image of the other half. To find it, imagine folding the shape along that line -- if the two halves match perfectly, it is a line of symmetry. For example, a square has 4 lines of symmetry.
How many lines of symmetry does a rectangle have compared to a square?
A rectangle has 2 lines of symmetry -- one going across the middle horizontally and one going down the middle vertically. A square has 4 lines of symmetry -- the same two as the rectangle, plus both diagonals. The diagonals do not work for a rectangle because the halves do not match.
What letters of the alphabet have a line of symmetry?
Letters like A, B, C, D, E, M, T, U, V, W, X and Y have at least one line of symmetry. For example, A has a vertical line of symmetry (left and right halves match). B and D have a horizontal line of symmetry (top and bottom halves match). Letters like F, G, J, P, Q, R, S and Z have no symmetry.
What is rotational symmetry and what does order of rotation mean?
A shape has rotational symmetry if you can rotate it around a central point and it looks exactly the same before it completes one full turn. The order of rotation is how many times the shape looks the same during one full 360° turn. For example, a square looks the same at 90°, 180°, 270° and 360° -- so its order of rotation is 4.
What is the angle of rotation for a shape with rotational symmetry?
The angle of rotation is the smallest angle you need to turn the shape for it to look exactly the same. You find it by dividing 360° by the order of rotation. For example, a regular hexagon has order 6, so its angle of rotation = 360° ÷ 6 = 60°.